 # Calculus Problems

## Transcribed Text

1. Sketch the parametric curve x = cost, y = cos(2t) (0 < t ≤ 2π), showing its direction. Find a Cartesian Equation in x and y whose graph contains the parametric curve. 2. Find the coordinates of the points at which the parametric curve x = t 2 , y = t 3 − 3t has a) a horizontal tangent b) a vertical tangent. 3. Find the Cartesian equation of the tangent to the parametric curve x = 3t + t 3 , y = t 3 − 9t 2 at the point t = 1. 4. For the curve x = t − sin t, y = 1 − cost, nd d 2y dx2 , and determine where the curve is concave up and concave down. 5. Find the length of the curve x = t − sin t, y = 1 − cost, 0 ≤ t ≤ 2π. Hint: to evaluate the integral use the identity sin2 θ = 1−cos(2θ) 2 . 6. Find the area of the surface generated by rotating the curve x = 3t − t 3 , y = 3t 2 , 0 ≤ t ≤ 1 about the x-axis. 7. Transform the polar equation r = 5 3−4 sin θ to rectangular coordinates. 8. Find the area of the region that lies inside the graphs of both polar equations r = 1 and r = 2 sin θ. 9. Find the length of the polar curve r = e −θ , 0 ≤ θ ≤ 3π. 10. Find the limit of the sequence an = ln(2+e n) 3n , if it exists. 11. Find the sum of the series P∞ n=0 3 4 n 5n+1 12. Calculate the sum of the series: X∞ n=1 1 n(n + 2) 13. Determine whether each of the following series converge or not. (Name the test you use. You do not have to evaluate the sums of those series.) a) P∞ n=1 √ n n3+1 b) P∞ n=2 1 n ln n c) P∞ n=1 n! nn . 14. Determine whether each of the following series converge absolutely, converge conditionally or diverge. 1 a) P∞ n=1 (−1)n n √ n b) P∞ n=0(−1)n−1 n n2+1 15. Starting with the geometric series 1 + x − x 2 + x 3 + . . . (−1 < x < 1), nd the power series for x (1−x) 2 in powers of x. a) Where is the series valid ? b) Using the result in (a) nd the sum P∞ n=1 n 2n 16. Determine the radius and interval of convergence of the series P∞ n=0 (2x−1)n 3n . 17. Find the Maclaurin series for f(x) = x 2 e 2x starting with a familiar series. For that values of x is the representation valid ? 18. Find the Taylor series representation of f(x) = ln x in powers of x − 1. Find the radius of convergence of this Taylor series.

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